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Item type:Publication, Access status: without files , An Alternative Approach to Probability in Quantum Information ScienceThis chapter provides a probabilistic framework for formulating classical probability theory, quantum probability, thermodynamics, diffusion, and the Wiener integral using a set of four axioms or principles. It explains everything that conventional quantum information theory and classical probability theory achieve. We want to emphasize that this framework is not an interpretation of quantum mechanics such as “Many-Worlds,” “Bohm’s Theory,” or the “Copenhagen interpretation.” It is much more general and can be viewed as a probability algorithm that calculates probabilities of future events. As a result, previously perplexing paradoxes find resolution. In particular, the superposition principle takes on a new meaning. Our probabilistic framework stands apart from the Hilbert space formalism. It relies solely on elementary set theory, classical logic, and complex numbers. Consequently, this theory is accessible for instruction in educational settings. This framework can be regarded as an axiomatic approach to probability in the sense of Hilbert. In his sixth of the twenty-three open problems presented at the International Congress of Mathematicians in Paris in 1900, Hilbert called for an axiomatic probability theory.Publicationtype: Book Part18 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Conceptual basis of probability and quantum information theory(2022-09-15)These notes present a probabilistic framework that enables a formulation of classical probability theory, thermodynamics, and quantum probability with a common set of four principles or axioms. It explains everything that usual quantum mechanics and classical probability theory does. We emphasize that this framework is not an interpretation of quantum mechanics, such as ''many worlds``, ''Kopenhagen interpretation``, or others. It is a probability algorithm that computes probabilities of future events and additionally enables a reconstruction of quantum theory, thermodynamics, diffusion, and Wiener processes. We distinguish strictly between possibilities and outcomes. Moreover, we use a time concept based on the classification of future, present, and past. Well-known paradoxes are resolved. The superposition principle obtains a new meaning. The inclusion-exclusion principle, well-known in probability theory and number theory, is generalized to complex numbers. Our probabilistic framework is not based on the Hilbert space formalism. It requires only simple set theory and complex numbers. Thus, this theory can be taught in schools. Our framework may be viewed as an axiomatic approach to probability in the sense of Hilbert, who asked for an axiomatic probability theory in his sixth of the twenty-three open problems presented to the International Congress of Mathematicians in Paris in 1900. We have applied our probabilistic algorithm to several problems, including classical problems, statistical mechanics and thermodynamics, diffraction at multiple slits, light reflection, interferometer, delayed-choice experiments, and Hardy's Paradox.Publicationtype: Working Paper106 366 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A Unified treatment of classical probability, thermodynamics, and quantum information theory(2021-09-14)The major goal of these notes is an elaborate presentation of a probabilistic framework. This framework allows a formulation of classical probability theory, thermodynamics, and quantum probability with a common set of four principles or axioms. In particular, it provides a general prognostic algorithm for computing probabilities about future events. Our principles distinguish strictly between possibilities and outcomes. A well-defined possibility space and a sample space of outcomes resolves well-known paradoxes, and make quantum interpretations like ''many worlds`` or ''many minds`` superfluous. In addition, the superposition principle and the entanglement of systems obtain a new meaning from our point of view. This framework offers an axiomatic approach to probability in the sense of Hilbert. He asked for treating probability axiomatically in his sixth of the twenty-three open problems presented to the International Congress of Mathematicians in Paris in 1900. We have applied our framework to various problems, including classical problems, statistical mechanics and thermodynamics, diffraction at multiple slits, light reflection, interferometer, delayed-choice experiments, and Hardy's Paradox. Particular emphasis is also placed on C.F. von Weizs\"acker's work, who developed his ur theory as early as the 1950s. Today, leading researchers continue his work under the name ''Simons Collaboration on Quantum Fields, Gravity, and Information``.Publicationtype: Book231 702 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Rigorous lower bounds for the ground state energy of molecules by employing necessary N-representability conditions(2020-11-13); ; ; ; Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced density matrix defines the variable. This concept leads to large-scale semidefinite programming problems that provide a lower bound for the ground state energy. Upper bounds of the ground state energy can be calculated for example with the Hartree–Fock method or numerically more exact for a given basis set by full CI. However, Nakata et al. ( J. Chem. Phys.200111482828292) observed that due to numerical errors the semidefinite solver produced erroneous results with a lower bound significantly larger than the full CI energy. For the LiH, CH–, NH–, OH, OH–, and HF molecules violations within one mhartree were observed. We applied the software VSDP which takes all numerical errors due to floating-point arithmetic operations into consideration. For two test libraries VSDP provides tight rigorous error bounds lower than full CI energies reported with an accuracy of 0.1 to 0.01 mhartree. Only little computation work must be spent in order to compute close rigorous error bounds for the ground state energy.Publicationtype: Journal Article97 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Quantum information theory for engineers : free climbing through physics and probability(2019-06-13)The aim of this supplement to my lecture notes ''Quantum Information Theory for Engineers: An Interpretative Approach`` is to develop a probabilistic framework that can be applied to classical stochastic problems as well as to quantum mechanical problems with interference phenomena. In particular, many paradoxes in quantum theory are demystified. Our approach replaces spacetime by a trinity of time, namely future, present, and past leading to a distinction between outcomes, possibilities, and internal possibilities. Our approach unifies the Dirac-Feynman rules with classical probability theory. We discuss polarization and slit experiments, the measurement problem, quantum electrodynamics, and Hardy's paradox, a spectacular experimental setup where simple logical arguments about its physical constraints lead to a surprising contradiction.Publicationtype: Book498 306 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Quantum information theory for engineers: an interpretative approach(2017)These lecture notes comprise lectures on quantum computing and quantum information theory that I taught during the last ten years for students of electrical engineering, mathematics, and computer science. The notes present an alternative entrance to quantum information theory that is suitable for students studying engineering, but perhaps also for people interested in physics and philosophy of physics.Publicationtype: Book1863 1214 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Rigorous results in electronic structure calculationsElectronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced density matrix defines the variable. This concept leads to large-scale semidefinite programming problems that provide a lower bound for the ground state energy. Upper bounds of the ground state energy can be calculated for example with the Hartree-Fock method. However, Nakata, Nakatsuji, Ehara, Fukuda, Nakata, and Fujisawa observed, that due to numerical errors the semidefinite solver produced erroneous results with a lower bound significantly larger than the Hartree-Fock upper bound. Violations within one mhartree were observed. We present here a method for solving electronic structure problems where all numerical errors are taken into consideration. In particular this method provides rigorous error bounds without violations as mentioned above.Publicationtype: Journal Article47 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , On verified numerical computations in convex programmingThis survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transformations to special types of convex problems and convex relaxations is included. The latter are important for handling and for reliability issues in global robust and combinatorial optimization. Some remarks on numerical experiences, including also large-scale and ill-posed problems, and software for verified computations concludes this survey.Publicationtype: Journal Article46 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , A classical interpretation of quantum mechanics and the measurement problem(2008-01-21)In this paper a didactic approach is described which immediately leads to an understanding of those postulates of quantum mechanics used most frequently in quantum computation. Moreover, an interpretation of quantum mechanics is presented which is motivated by retaining the point of view of classical mechanics as much as possible, and which is consistent with relativity theory. Everything can be written down in terms of well-known mathematical formulations that can be found in every textbook about quantum mechanics. Therefore, in this version, almost no formulas are used.Publicationtype: Journal Article31 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Rigorous error bounds for the optimal value in semidefinite programming(Society for Industrial & Applied Mathematics (SIAM), 2007-12-01); ; A wide variety of problems in global optimization, combinatorial optimization, as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library; these libraries contain many ill-conditioned and real-life problems.Publicationtype: Journal Article43